Abstract
The inertia matrix of any rigid body is the same as the inertia matrix of some system of four point-masses. In this work, the possible disposition of these point-masses is investigated. It is found that every system of possible point-masses with the same inertia matrix can be parameterised by the elements of the orthogonal group in four-dimensional modulo-permutation of the points. It is shown that given a fixed inertia matrix, it is possible to find a system of point-masses with the same inertia matrix but where one of the points is located at some arbitrary point. It is also possible to place two point-masses on an arbitrary line or three of the points on an arbitrary plane. The possibility of placing some of the point-masses at infinity is also investigated. Applications of these ideas to rigid body dynamics are considered. The equation of motion for a rigid body is derived in terms of a system of four point-masses. These turn out to be very simple when written in a 6-vector notation.
Highlights
Two systems of rigidly connected point-masses are said to be equimomental if their inertia about any line in space is equal
Most articles refer to the treatise by Routh, “Dynamics of Rigid Bodies” [9]
Where m is the mass of the body; I is usual 3×3 inertia matrix of the body; I3 is the 3×3 identity matrix and C is an antisymmetric matrix corresponding to the position vector of the centre of mass
Summary
Two systems of rigidly connected point-masses are said to be equimomental if their inertia about any line in space is equal. The original idea seems to be due to Reye in a paper published in 1865 [10] In these works, several other theorems about the disposition of systems of point-masses equimomental to a body are given, in particular various ellipsoids that the points can lie on are described. The space of all possible four point-mass systems equimomental to a given body is identified as the quotient of the Lie group O(4) by the symmetric group of permutations on four letters. This result is believed to be novel, given the size and age of the literature on the subject it is difficult to be certain of this. This six-component vector form of the equations of motion is believed to be novel and may have applications to the design and analysis of mechanisms and robots
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